# WORKSHEET: Position and Velocity Functions

The purpose of this worksheet is to give you practice graphing position functions, and to practice the concept of velocity.

• For each story below, create a graph that matches the story.
• All the stories have the same set-up:
you live on a long, straight, road (i.e., a number line).
Your house is at position $0$.
You always leave your house and turn right, which is the positive direction (i.e., towards $1, 2, ...$).
• Let $p(t)$ denote the position of the car at time $t$;
put $p(t)$ along the vertical axis, and $t$ along the horizontal axis.
• Clearly label the times that are indicated in the stories ($a < b < c$ etc.) on your graphs.
You may choose the locations of these times!
• When you are done with the the graphs, answer the questions that follow.
1. Leave your house at $t = 0$. Gradually speed up so that you are going $50$ mph at $t = a$.
2. Drive at a constant speed until $t = b$.
3. Gradually slow down until you reach $30$ mph at $t = c$.
4. Drive at a constant speed until $t = d$.
5. Come quickly to a stop at $t = e$.
6. Back up the car at a constant speed of $10$ mph until $t = f$.
7. Go forward again until $t = g$.
1. ‘the Police Car’
1. Leave your house at $t = 0$. Gradually speed up to $70$ mph at $t = a$.
2. See a police car ahead of you on the road. Step on the brakes at $t = b$.
3. Manage to get down to $50$ mph (the speed limit) by the time you pass the police car at $t = c$.
4. After the police car is out of sight, gradually increase your speed so that you are going $70$ mph again at $t = d$.
5. While you're thinking “Whoa. Close call!” you see another police car—too late! The flashing lights go on. The police car pulls you over, so you slow down quickly; at $t = e$ you pull over to the side of the road.
6. You remain at the side of the road while the police person writes out a ticket. At $t = f$, you say good-bye and slowly pull back into traffic.
7. You accelerate to $50$ mph for the rest of the trip, until you start slowing down at $t = g$ to get ready to stop.
8. You are at a complete stop at $t = h$.
2. ‘I always forget things!’
1. Leave your house at $t = 0$. You're really tired today and bummed out because of yesterday's speeding ticket, so you gradually speed up to $40$ mph at $t = a$.
2. Drive at a constant speed with your thoughts wandering until $t = b$.
3. At $t = c$, you suddenly realize that you've forgotten a homework assignment that needs to be passed in today. So, you quickly come to a complete stop by $t = d$.
4. You turn around, and speed up to $50$ mph by $t = e$.
5. You drive at $50$ mph until you start to slow down for your driveway at $t = f$.
6. You slow down, coming to a complete stop at $t = g$.
7. You run into your house and grab your homework. While there, the phone rings, which takes a few more minutes. Then, you get back out to your car at $t = h$ and take off again.
8. You're now late, so you quickly speed up to $55$ mph by $t = i$.
9. You drive at $55$ mph until you slow down and come to a complete stop at your final destination by $t = f$.